climxrrelem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > climxrrelem		Structured version Visualization version GIF version

Theorem climxrrelem 45042

Description: If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

**Hypotheses**
Ref	Expression
climxrrelem.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
climxrrelem.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climxrrelem.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
climxrrelem.c	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climxrrelem.d	⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
climxrrelem.p	⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
climxrrelem.n	⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))

**Assertion**
Ref	Expression
climxrrelem	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)

Distinct variable groups: 𝐴,𝑗 𝐷,𝑗 𝑗,𝐹 𝑗,𝑀 𝑗,𝑍 𝜑,𝑗

Proof of Theorem climxrrelem Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1909	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfv 1909	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
3		nfra1 3275	. . . . . 6 ⊢ Ⅎ𝑘∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
4	2, 3	nfan 1894	. . . . 5 ⊢ Ⅎ𝑘(𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
5	1, 4	nfan 1894	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
6		climxrrelem.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_≥‘𝑀)
7	6	uztrn2 12845	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
8	7	adantll 711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
9		climxrrelem.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
10	9	fdmd 6722	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝑍)
11	10	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → dom 𝐹 = 𝑍)
12	8, 11	eleqtrrd 2830	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
13	12	adantlrr 718	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ dom 𝐹)
14		simpll 764	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
15	8	adantlrr 718	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
16		rspa 3239	. . . . . . . 8 ⊢ ((∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
17	16	adantll 711	. . . . . . 7 ⊢ (((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
18	17	adantll 711	. . . . . 6 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
19	9	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ^*)
20	19	3adant3 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ^*)
21		simpll 764	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝜑)
22		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) = -∞)
23		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → (𝐹‘𝑘) ∈ ℂ)
24	22, 23	eqeltrrd 2828	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = -∞) → -∞ ∈ ℂ)
25	24	adantll 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → -∞ ∈ ℂ)
26		climxrrelem.n	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ -∞ ∈ ℂ) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
27	21, 25, 26	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
28	27	adantlrr 718	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ≤ (abs‘(-∞ − 𝐴)))
29		fvoveq1 7428	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = -∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴)))
30	29	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(-∞ − 𝐴)))
31		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
32	30, 31	eqbrtrrd 5165	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
33	32	adantll 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
34	33	adantlrl 717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) < 𝐷)
35		climxrrelem.c	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
36	6	fvexi 6899	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑍 ∈ V
37	36	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑍 ∈ V)
38	9, 37	fexd 7224	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ V)
39		eqidd 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝐹‘𝑘) = (𝐹‘𝑘))
40	38, 39	clim 15444	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹 ⇝ 𝐴 ↔ (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))))
41	35, 40	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥)))
42	41	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℂ)
43	42	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → 𝐴 ∈ ℂ)
44	25, 43	subcld 11575	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (-∞ − 𝐴) ∈ ℂ)
45	44	abscld 15389	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
46	45	adantlrr 718	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → (abs‘(-∞ − 𝐴)) ∈ ℝ)
47		climxrrelem.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
48	47	rpred 13022	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
49	48	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → 𝐷 ∈ ℝ)
50	46, 49	ltnled 11365	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ((abs‘(-∞ − 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴))))
51	34, 50	mpbid 231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = -∞) → ¬ 𝐷 ≤ (abs‘(-∞ − 𝐴)))
52	28, 51	pm2.65da 814	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞)
53	52	3adant2 1128	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = -∞)
54	53	neqned 2941	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ -∞)
55		simpll 764	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝜑)
56		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) = +∞)
57		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → (𝐹‘𝑘) ∈ ℂ)
58	56, 57	eqeltrrd 2828	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑘) ∈ ℂ ∧ (𝐹‘𝑘) = +∞) → +∞ ∈ ℂ)
59	58	adantll 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → +∞ ∈ ℂ)
60		climxrrelem.p	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ℂ) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
61	55, 59, 60	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
62	61	adantlrr 718	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ≤ (abs‘(+∞ − 𝐴)))
63		fvoveq1 7428	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑘) = +∞ → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴)))
64	63	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) = (abs‘(+∞ − 𝐴)))
65		simpl 482	. . . . . . . . . . . . . 14 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)
66	64, 65	eqbrtrrd 5165	. . . . . . . . . . . . 13 ⊢ (((abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷 ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
67	66	adantll 711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
68	67	adantlrl 717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) < 𝐷)
69	42	ad2antrr 723	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → 𝐴 ∈ ℂ)
70	59, 69	subcld 11575	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (+∞ − 𝐴) ∈ ℂ)
71	70	abscld 15389	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝐹‘𝑘) ∈ ℂ) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
72	71	adantlrr 718	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → (abs‘(+∞ − 𝐴)) ∈ ℝ)
73	48	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → 𝐷 ∈ ℝ)
74	72, 73	ltnled 11365	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ((abs‘(+∞ − 𝐴)) < 𝐷 ↔ ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴))))
75	68, 74	mpbid 231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) ∧ (𝐹‘𝑘) = +∞) → ¬ 𝐷 ≤ (abs‘(+∞ − 𝐴)))
76	62, 75	pm2.65da 814	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞)
77	76	3adant2 1128	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → ¬ (𝐹‘𝑘) = +∞)
78	77	neqned 2941	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ≠ +∞)
79	20, 54, 78	xrred 44652	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍 ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)) → (𝐹‘𝑘) ∈ ℝ)
80	14, 15, 18, 79	syl3anc 1368	. . . . 5 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
81	13, 80	jca 511	. . . 4 ⊢ (((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
82	5, 81	ralrimia 3249	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))
83	9	ffund 6715	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
84		ffvresb 7120	. . . . 5 ⊢ (Fun 𝐹 → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
85	83, 84	syl 17	. . . 4 ⊢ (𝜑 → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
86	85	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → ((𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)))
87	82, 86	mpbird 257	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
88		breq2 5145	. . . . . 6 ⊢ (𝑥 = 𝐷 → ((abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥 ↔ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
89	88	anbi2d 628	. . . . 5 ⊢ (𝑥 = 𝐷 → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
90	89	rexralbidv 3214	. . . 4 ⊢ (𝑥 = 𝐷 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
91	41	simprd 495	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑥))
92	90, 91, 47	rspcdva 3607	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
93		climxrrelem.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
94	6	rexuz3 15301	. . . 4 ⊢ (𝑀 ∈ ℤ → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
95	93, 94	syl 17	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷) ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷)))
96	92, 95	mpbird 257	. 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝐷))
97	87, 96	reximddv 3165	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 Vcvv 3468 class class class wbr 5141 dom cdm 5669 ↾ cres 5671 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 ℝcr 11111 +∞cpnf 11249 -∞cmnf 11250 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12980 abscabs 15187 ⇝ cli 15434

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438

This theorem is referenced by: climxrre 45043

W3C validator